In this study we consider a wave energy farm consisting of a large number of bottom-hinged paddles. The paddles are arranged in a doubly-periodic manner, with a finite number of identical rows and each row containing an infinite number of paddles. Each paddle is attached to its own damper and spring, allowing power to be generated through its pitching motion. Unlike previous studies into paddle arrays, we envisage compact arrays consisting of small devices arranged across a large number of rows. The mathematical analysis of this proposed configuration is best suited to methods that exploit the periodicity of the array. It is shown that the velocity potential everywhere within the array can be described by an expansion in terms of suitably-defined Floquet-Bloch eigenmodes and this allows the solution to the scattering problem involving waves incident upon the array to be determined by a simple low-order system of equations. The method used in this study is exact within the setting of linearised water waves and has all of the advantages of classical low-frequency multi-scale homogenization without the low-frequency restriction. It may also be regarded as a natural extension of the well-known wide-spacing approximation without the large separation restriction. Although the focus of this paper is on the mathematical approach to the solution of the problem, we provide a range of results to illustrate the performance of this proposed wave energy converter concept.